Temperature Control Valve

Not necessarily. Among various things, it depends on whether the valve is in a diverting or blending configuration. For instance, if it is a blending valve, it could receive two streams at different temperatures and pressures, mixing the two streams into an outlet stream, with temperature a weighted average of the inlet temperatures, and pressure less than either inlet stream pressure. A P&ID would help in clarifying this.

In that case, the two exit streams should be at the same temperature as the inlet stream (or very very slightly cooler); the exit pressures will be less than the inlet pressure according to the flows and the valve's C_v in each direction.