I have no heat to the heater core in the truck. There is a Tee before the firewall and one side is hot and the other is cold, there is no valve at that location

Most likely it is the heater control valve that shifts air from blowing across your heater core to your air conditioner core.

If your fan works and your car is fairly new..meaning your heater core isn't dirty and full of craap...then it is the climate cotrol valve.
If you are lucky, it might be a thermostat that is bad and blocking hot water from entering your heater core. Bottom line, if hot water doesn't circulate thru that little heater core, you won't get heat.
You might be able to look under the hood and grab the hot water lines that go thru the firewall and feed your heater core. If they feel cold to the touch after engine has warmed up, then you aren't getting circulating water to the heater core. If they feel nice and warm, then the heater core is fine, but your climate control valve has closed off the fan that blows across the core to disperse heat into the cabin

1: not enough liquid in the system (air in the heater core)
2: inoperative heater valve not allowing hot liquid into the core
3: hot/cold blend door in the heater box stuck on cold
4: clogged heater core
A clogged heater core will have one hot hose and one significantly cooler hose at the firewall connection; easily tested by hand. a good core will have both hoses very hot to the touch when the heater is on full blast.
Make sure that the system is full and at operating temperature, and the heater controls set on full heat. Feel the hoses at the firewall; if they are both too hot to touch then the problem is in the heater box or controls.

Hello! Thermostat should be a 190 degree...you may have an air blockage...Remove radiator cap start engine and leave cap of until all air bubbles are out (even after thermostat opens)...Cap and test for heat...if there isn't any heat...When that thermostat opens and hot water flows through the upper hose of the radiator that hose will be hot and hard..If you feel the heater hose only one will be hot which means that nice hot water is not passing through the heater core...That could be due to blockage or the control in the car isn't opening the valve on the core...check to see if the heat control is opening the valve...If it is, I would suggest that you purchase a back flushing kit ($10) A plastic tee fits into the inlet heater hose and a garden hose is then connected to the tee...Drain the radiator, remove the thermostat, and turn on the water...do this with heat control on high...The system will back flush out of the top of the radiator and hopefully clean the blockage...Reassemble ...Cap off the tee (cap is in kit)...Install thermostat (190) Is a better choice...Refill radiator...Start engine and leave the radiator cap off until all the air bubbles out............At that point you should have heat...Guru...Saailer

I’m happy to help further over the phone at https://www.6ya.com/expert/david_6df67de3b14de867

With the engine warm and the heat turned on, feel the hoses going through the firewall to the heater core. If one is hot and the other cold, you have a clogged heater core which must be backflushed to clear a blockage.

If the hoses are both cold, move up the hoses until you find a valve. This is the heater control valve. If it is hot on one side of the valve, that valve is clogged or shut. Investigate why the valve is not passing coolant - a clog or the actuator is broken/disconnected.

If both sides of the heater control valve are cold, replace the thermostat. While you have the thermostat housing open, make sure there are no blockages in the connection pipes.

Check to see that the blend door linkage is hooked up. Doubtful it is heater core, as you had heat to one side and not the other. Check coolant control valve linkage, verify it is moving the coolant valve from off to on when you switch temperature from cold to hot. Verify you have coolant flow to the heater core by feeling the heater hoses where they enter the firewall. If all checks out well, then look at hvac controls individually.

sounds like a restriction in the coolant bypass and or heater core. you will want to check if both sides of the heater core hose near the firewall are hot on both sides if so you have an issue with the air ducts but if not then you may have a failing heater control valve or a restriction.

If you look by the firewall there are two hoses. They go into the heater core. When the engine is warmed up, and the heater on. Feel if both hose are equally warm. Be carefull you don't burn yourself. A valve lets the hot water circulate through the heater core therefore providing heat. Also feel if the heater core is warm under you dash on the passenger side. If it feels cold then water is not circulating. Locate the valve, which is usually vacuum operated. See if there is vacuum there when the heater is in the on position.

Sounds like a stuck heater core valve. Find heater core valve. Under hood, find the two heater core hoses going into firewall. One of the hoses has a valve. Operate temperature control (full heat, then full cold) check that valve changes position. If not either replace valve or control system failure. Also, when temperature control in cold position, one hose should feel hot and the other cold. Temp fix: disable valve to close no heat or on heat.

Under the hood on the pass. side firewall are two heater hose lines. Start the car and let it get at normal operating temp. One of the hoses is the "in" to the heater core, if you touch it it should be hot. The other hose is the "out" of the heater core, this hose should be warm and not really hot like the "in" hose. If the hose going in is hot and the other cold, then your heater core is clogged. There is a valve connected inline with one of the heater hoses, this is the heater control vavle.
If the valve does not open, you will not have any heat either.