Let $\mathcal C$ be a category whose skeleton has $\lambda$-many objects and $\kappa$-many morphisms. Then the skeleton of the endofunctor category $\mathcal C^{\mathcal C}$ has at most $\kappa^\lambda$ many morphisms.

My guess is that in most cases, this upper bound is achieved.

Question: What is an example of a category $\mathcal C$ (other than the terminal category or equivalents) whose skeleton has $\lambda$ many objects and $\kappa$ many morphisms, but the skeleton of whose endofunctor category $\mathcal C^{\mathcal C}$ has strictly fewer than $\kappa^\lambda$ many morphisms? Can it have as few as $\kappa$-many morphisms and $\lambda$-many objects?

Note that $\mathcal C^{\mathcal C}$ always has (skeletally) at least $\kappa+1$-many morphisms, given by constant morphisms between constant endofunctors, along with the identity functor.