dim as a functor

Let $\mathbb{F}-{\mathrm{V}}_{\mathrm{B}}$$\mathbb F-\mathrm{V_B}$ be the category of finite dimension vector spaces with basis.

Let $\mathbb{N}$$\mathbb N$ be a category. The object is $n=\{1,2,...,n\}$$n=\{1,2,...,n\}$. The morphism are functions between them.

Then $\dim$$\dim$ is a functor.

For the object, $(V,B)$$(V,B)$, the $\dim$$\dim$ map it to $(V,\{b_1,b_2,...,b_n\})\to n$$(V,\{b_1,b_2,...,b_n\})\to n$

For the morphism, $T$$T$ is determined at the basis. Hence $\dim (T)(b_i)=T(b_i)$$\mathrm{dim}(T)(b_i)=T(b_i)$ gives you the function between $n$$n$ and $m$$m$.

Hence $\dim$$\dim$ is functor.