Given a finite dimensional $k$-algebra $A$ and two (finitely generated) left $A$-modules $M, N$ we can view $\text{Hom}_A(M,N)$ as a subset of the k-linear maps $\text{Hom}_k (M,N)$.
Now also consider a f.d. $k$-algebra $B$ and a covariant additive functor $$ F \colon {B~\text{mod}} \to {A~\text{mod}}.$$ We can view $F(B)$ as a left $A$ - right $B$ -bimodule. Does a similar description to the above then hold for the left $B$-module $\text{Hom}_A\left(F(B),N\right)$?