Given functions $f\colon X\to Y$ and $g\colon Y \to Z $, their composition, that is the function $h\colon X\to Z, x \mapsto g(f(x))$ is denoted by $h=g\circ f$. Let's call this the functional form, since it does not involve the variable $x$.
Consider the case of multivariate functions. Given $f_1\colon X_1\to Y_1$, $f_2\colon X_2 \to Y_2$ and $g\colon Y_1 \times Y_2 \to Z$ is there an established notation to denote partial function compositions such as $h\colon X_1 \times Y_2 \to Z, (x,y) \mapsto g(f_1(x),y)$ in functional form?
Best Answer
The function $h$ that you prescribe can be denoted as:$$g\circ(f_1\times\mathsf{id_{Y_2}})$$where $f_1\times\mathsf{id_{Y_2}}:X_1\times Y_2\to Y_1\times Y_2$ is prescribed by $(x,y)\mapsto(f_1(x),y)$ and $\mathsf{id_{Y_2}}$ denotes the identity map on $Y_2$.