Northcott Multilinear Algebra Universal Property Proof Northcott Multilinear Algebra poses a pro

Northcott Multilinear Algebra Universal Property Proof
Northcott Multilinear Algebra poses a problem. Consider R-modules ${\displaystyle M_1,\dots ,M_p}$, M and N. Consider multilinear mapping
${\displaystyle \psi :M_1\times \cdots \times M_p\to N}$
Northcott calls the universal problem as the problem to find M and multilinear mapping ${\displaystyle \varphi :M_1\times \cdots \times M_p\to M}$ such that there is exactly one R-module homomorphism ${\displaystyle h:M\to N}$ such that ${\displaystyle h\circ \varphi =\psi }$.
If ${\displaystyle \lambda \text{and}{\lambda }^{\prime }}$ exist I understand why the equalities at the end of the sentence follow, based on the satisfaction of the universal problem. I can't see however why homomorphisms ${\displaystyle \lambda \text{and}{\lambda }^{\prime }}$ should exist.
I did more group theory many years ago and this is my first serious foray into "modules" so I wouldn't be surprised if there is something obvious I'm missing.
my thoughts: Clearly M and M′ are both homomorphic to N through h and h′, I'm not sure if this says anything about a relationship between M and M′ though.
If h′ were injective I could say something like ${\displaystyle \lambda \left(m\right)=h^{{}^{\prime }-1}\left(h\left(m\right)\right)}$ but I don't know if there is any guarantee that h′ is injective..
Likewise, if ${\displaystyle \varphi }$ were injective I could define ${\displaystyle \lambda \left(m\right)={\varphi }^{\prime }\left({\varphi }^{-1}\left(m\right)\right)}$ but again I don't know why this would be the case...
I've tried replacing M and N with more familiar vector spaces and R-module homomorphisms by multilinear maps for better intuition but no luck.. I do know that if M and M′ are vector spaces with the same dimension then there is an isomorphism between them. I guess more generally if M and M′ have different dimensions (say ${\displaystyle \dim \left(M^{\prime }\right)>\dim \left(M\right))}$ then there is a homomorphism from M into a subspace of M′ and another homormophism from M′ onto M. Maybe this carries over to modules and is in the right direction for what I need...?