Matrix Coefficients Map a Coalgebra Morphism

Question

Simple question: let $C$ be a coalgebra over a field $k$. Given a finite-dimensional right $C$-comodule $V$ determined by the structure morphism $$a:V\to V\otimes C,$$ we have a natural 'matrix coefficients' morphism given by $$V^*\otimes V\xrightarrow{1\otimes a}V^*\otimes V\otimes C\xrightarrow{ev\otimes 1}C.$$ Now $V^*\otimes V=End(V)^*$ has the natural structure of a coalgebra as the dual of the algebra $End(V)$, so we can ask if this morphism is one of coalgebras. And that is precisely my question! I would appreciate an answer and/or reference. Thanks :)