Is there a "medium" \cap?

You can use scalebox from the graphicx package to create one:

\documentclass{article}
\usepackage{graphicx}

\newcommand*{\medcap}{\mathbin{\scalebox{1.5}{\ensuremath{\cap}}}}%
\begin{document}
\[ Hom(A,B) \cap    Hom(B,C)\]
\[ Hom(A,B) \medcap Hom(B,C)\]
\[ Hom(A,B) \bigcap Hom(B,C)\]
\end{document}

I don't find it to be too small. You can use the same symbol as \bigcap, but not in the "bigger" version, so there's some work to do:

\newcommand*\mcap{\mathbin{\mathpalette\mcapinn\relax}}
\newcommand*\mcapinn[2]{\vcenter{\hbox{$\mathsurround=0pt
  \ifx\displaystyle#1\textstyle\else#1\fi\bigcap$}}}

This is the result of

\[\operatorname{Hom}(A,B)\cap\operatorname{Hom}(B,C)\]

and of

\[\operatorname{Hom}(A,B)\mcap\operatorname{Hom}(B,C)\]

You need \usepackage{amsmath} to use \operatorname and you might want to say

\DeclareMathOperator{\Hom}{Hom}

in the preamble, in order to be able to use \Hom as an abbreviation.

You can use scalerel from the scalerel package to create one:

\documentclass{article}
\usepackage{scalerel}

\begin{document}
\[ Hom(A,B) \cap Hom(B,C) \]
\[ Hom(A,B) \scalerel{\cap}{Hom(B,C)} \]
\[ Hom(A,B) \mathbin{\scaleobj{1.75}{\cap}} Hom(B,C) \]
\[ Hom(A,B) \mathbin{\scaleobj{0.9}{\bigcap}} Hom(B,C) \]
\[ Hom(A,B) \bigcap Hom(B,C) \]
\end{document}