# The category $\bf{FinVect}$ of finite vector spaces is rigid.

I am following Pavel Etingof et al's book on tensor categories.

They give FinVect as an example of a rigid monoidal category, with evaluation map given by $$\text {ev}_V(\epsilon\otimes v)=\epsilon(v)$$ and coevaluation map $$\text{coev}_V:\mathbb K \rightarrow V\otimes V^*$$ "the usual embedding".

My question is embarassingly simple: what is this "usual embedding?" I was trying to see what it had to be using the definition of evaluation and coevaluation, but got nowhere.

What Etingof et al call the "usual embedding" is given by linear extension of the map$$1_K\mapsto\sum_{i = 1}^{n}v_i\otimes v_i^*$$
where the $$v_i$$ are a basis of V and $$v_i^*$$ denoted the basis dual to the $$v_i$$'s.
This makes perfect sense if you want $$ev \circ f\otimes id\circ coev = Trace(f)$$ for any $$f:V\longrightarrow V$$.
Try $$f=id$$ and you'll find that $$ev\circ coev = dim(V)$$
• To rephrase things slightly, if $V$ is finite dimensional, there is an isomorphism with $\hom(V,V)$ and $V\otimes V^{*}$, and the usual embedding is the inclusion of scalar maps into all linear maps. – Aaron Dec 4 '18 at 11:52
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