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Here is an interesting question from "The Rising Sea" book of Algebraic Geometry: http://math.stanford.edu/~vakil/216blog/FOAGapr2915public.pdf

1.2.3 Example. Consider the category $Vec_K$ of vector spaces over a field $K$. The objects are $K$-vector spaces, and the morphisms are linear transformations. What are the isomorphisms?

I would say that the isomorphisms are the invertible linear transformations, is that correct?

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    $\begingroup$ Yes, that's right. There's not much more to say about this, except to remark that it suffices to require that the linear transformation is invertible as a map of sets; this kind of thing isn't always true in other concrete categories. $\endgroup$ Dec 28, 2020 at 6:54

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Question: "I would say that the isomorphisms are the invertible linear transformations, is that correct?"

Answer: If $C$ is any category and $U,V$ objects in $C$, we define $U$ to be isomorphic to $V$ (written $U \cong V$) iff there are morphisms (in $C$) $f:U \rightarrow V$ and $g: V\rightarrow U$ such that $f\circ g=Id_V$ and $g \circ f=Id_U$. In your case the maps $f,g$ are invertible linear maps/linear transformations between vector spaces.

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    $\begingroup$ It's just the definition of isomorphism in a general category. $\endgroup$
    – Berci
    Dec 28, 2020 at 10:31

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