Showing that the direct product does not satisfy the universal property of the direct sum

I feel intuitively that for $\prod_{i\in N}\mathbb{Z}$, as a $\mathbb{Z}$−module, and $\phi_i:\mathbb{Z}\to\mathbb{Z}$ the identity map, more than one homomorphism $\phi:\prod_{i\in N}\mathbb{Z}\to\mathbb{Z}$ satisfies the condition “composition with the canonical embedding yields the identity map $\phi_i:\mathbb{Z}\to\mathbb{Z}$”. But I can’t myself define such homomorphisms $\phi:\prod_{i\in N}\mathbb{Z}\to\mathbb{Z}$.

Could anyone suggest some appropriate functions?

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