Multilinear Morphism of Monoids

A function f : ∏ i : I X i → Y : 𝑓 → subscript product : 𝑖 𝐼 subscript 𝑋 𝑖 𝑌 f:\prod_{i:I}X_{i}\to Y from an indexed product of Monoids X i subscript 𝑋 𝑖 X_{i} to Y 𝑌 Y is multilinear if it is a Monoid Homomorphism "separately" in each variable.

Explicitly, this means that for all i : I : 𝑖 𝐼 i:I and a family of elements x j : i ≠ j → X j : subscript 𝑥 𝑗 𝑖 𝑗 → subscript 𝑋 𝑗 x_{j}:i\neq j\to X_{j} , the composition f ∘ x j : X i → Y : 𝑓 subscript 𝑥 𝑗 → subscript 𝑋 𝑖 𝑌 f\circ x_{j}:X_{i}\to Y is a Monoid Homomorphism.

Questions

Can I make this displayed? I could consider a family of functions f : ( i : I ) → X i → Y u ⁢ ( i ) f:(i:I)\to X_{i}\to Y_{u(i)} as a sort of multi-multilinear map?