What is the name of this type of function composition?
You composition is of type $\operatorname{comp}:(A \to X \to C)\to (B \to X) \to (A\to B \to C)$. Up to Currying, it can be seen as being of type $(A  \times X \to C)\to (B \to X) \to (A\times B \to C)$. So it's just a composition of a two arguments functions with a one argument function. You take $f$ and $g$ and return $(a,b)\mapsto f(a,g(b))$.

From a general abstract nonsense point of view, you can define the functor $$A\times -:\begin{array}{lll}
O &amp; \mapsto &amp;A\times O\\
g : B \to X&amp;\mapsto &amp;&lt;\operatorname{id}_A;g&gt;:A\times B \to A\times X
\end{array}$$
and then your composition is just $\operatorname{comp}=f\mapsto g \mapsto f\circ (A\times  g)$.