What are the isomorphisms in the category of vector spaces, $Vec_K$? Here is an interesting question from &quot;The Rising Sea&quot; 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?
Question: &quot;I would say that the isomorphisms are the invertible linear transformations, is that correct?&quot;
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.
Question: &quot;I would say that the isomorphisms are the invertible linear transformations, is that correct?&quot;
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.