I am wondering what the explicit definition of the internal hom functor $[X,\_]$ (described in this answer), for the category of vector spaces $\text{Vect}$ (over, say, the field of real numbers) is.
I am guessing that $\text{Vect} \overset{[X,\_]}{\longrightarrow} \text{Vect}.$ I am guessing that for a vector space $Y$ we have that $[X,Y]$ has elements as linear maps $\phi$ from $X$ to $Y,$ and $X\overset{\phi + \phi'}{\rightarrow}Y$ sends $x \in X$ to $\phi(x)+\phi'(x)\in Y.$ Also, I guess for a scaler $\lambda,$ we have $X\overset{\lambda. \phi }{\rightarrow}Y$ sends $x \in X$ to $\lambda. \phi(x)\in Y.$
Is the above correct ? If not what should it be ?
The main thing I am wondering is how the functor $[X,\_]$ works on arrows. In other words, if I have a linear map $A \overset{\psi}{\rightarrow} B$ then how is the linear map $[X,A] \overset{[X,\psi]}{\longrightarrow} [X,B]$ defined ?
It would be really great if the answer could be as explicit and simple as possible (i.e., avoiding mention of tensors, adjunctions, modules etc.) as I am not well versed in linear or abstract algebra.
Best Answer
yes, this is correct.
The answer really depends on your background, at this point; if you want an intuitive argument, the functor $[X,\_]$ acts as post-composition by $\psi$; this is a map of vector spaces, if you define the structure as above: for the sake of brevity, let's call $[X,\psi]=\psi_*$ (this is a common notation); then, $\psi_*(\alpha) = \psi\circ\alpha$, and (since this equation is true elementwise) $\psi_*(\alpha+\beta)=\psi_*(\alpha) + \psi_*(\beta)$, and similarly with scalar multiplication.
A more elaborate, conceptual argument requires what you want to avoid: adjunctions between categories.