Evaluating characterizations of homomorphisms on truncated vector lattices of functions

Let $L$ be a (non necessarily unital) truncated vector lattice of real-valued functions on a nonempty set $X$. A nonzero linear functional $\psi$ on $L$ is called a truncation homomorphism if it preserves truncation, i.e.,% $$\psi\left( f\wedge\mathbf{1}_{X}\right) =\min\left\{ \psi\left( f\right) ,1\right\} \text{ for all }f\in L.$$ We prove that a linear functional $\psi$ on $L$ is a truncation homomorphism if and only if $\psi$ is a lattice homomorphism and% $$\sup\left\{ \psi\left( f\right) :f\leq\mathbf{1}_{X}\right\} =1.$$ This allows us to prove different evaluating characterizations of truncation homomorphisms. In this regard, a special attention is paid to the continuous case and various results from the existing literature are generalized.