Is this category essentially small? Let $\mathcal C$ be the category of finite dimensional $\mathbb C$-vector spaces $(V, \phi_V)$ where $\phi_V \colon V \to V$ is a linear map. A morphism $f \colon (V , \phi_V) \to (W , \phi_W)$ in this category is a linear map such that $\phi_W f = \phi_V f$. Note this category is the same as the category of $\mathbb C [t]$-modules whose underlying space is finite dimensional as a $\mathbb C$-vector space.
I am having some trouble working out how many isomorphism classes there are. The problem is that even if $V \cong W$ as vector spaces, the isomorphism might not respect the structure morphisms in $\mathcal C$. So potentially there are a LOT of isomorphism classes.

Jordan normal form tells you what the isomorphism classes look like, but you don't need to know this: it suffices to show that the collection of isomorphism classes with a fixed value of $\dim V$ forms a set, and this is straightforward as specifying the corresponding $\phi_V$ requires at most $(\dim V)^2$ parameters. 
