If standard function composition is defined as:

 (define compose
   { (B → C) → (A → B) 
             → (A → C) }

   F G -> (λ X (F (G X))))

What type of composition does the below function describe? Is there a particular name for it in category theory?

 (define compose-2
   { (A → (X → C)) → (B → X)
                   → (A → B → C) }

   F G -> (λ Y (compose (F Y) G)))

composition diagram


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 $$A\times -:\begin{array}{lll} O & \mapsto &A\times O\\ g : B \to X&\mapsto &<\operatorname{id}_A;g>:A\times B \to A\times X \end{array}$$ and then your composition is just $\operatorname{comp}=f\mapsto g \mapsto f\circ (A\times g)$.

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