In light of Yoneda's Lemma, it is useful to revisit Universal Properties I - Inspiring Examples to see how each can be framed in the same language: as a natural isomorphism between a hom-functor and another $\mathbf{\text{Set}}$-valued functor.

Examples in $\mathbf{\text{Set}}$

The Cartesian product of two sets

If $X$ and $Y$ are sets, then there is a natural bijection between the set of maps $Z\to X\times Y$, and the set of pairs of maps $Z\to X$, $Z\to Y$. In other words, there is a natural bijection

This bijection requires the two projection maps ${\pi}_{1}:X\times Y\to X$ and ${\pi}_{2}:X\times Y\to Y$ defined by $(x,y)\mapsto x$ and $(x,y)\mapsto y$, respectively. The bijection ${\varphi}_{Z}$ above is defined by sending each set map $f:Z\to X\times Y$ to the pair of set maps ${\pi}_{1}\circ f:Z\to X$, ${\pi}_{2}\circ f:Z\to Y$.

Examples in $\mathbf{\text{Ab}}$

The direct sum of two abelian groups

If $A$ and $B$ are abelian groups, their direct sum is the abelian group $A\oplus B$. As a set, it consists of all pairs of formal sums $a+b$ with $a\in A$ and $b\in B$. The operation is defined "component-wise": $(a+b)+({a}^{\prime}+{b}^{\prime})=(a+{a}^{\prime})+(b+{b}^{\prime})$. (Although not common, one could reasonably argue that a different notation should be used for the formal sum symbol, such as $a\oplus b$.) One can verify that $A\oplus B$ is an abelian group, and that it comes equipped with two injective group morphisms ${i}_{1}:A\to A\oplus B$ and ${i}_{2}:B\to A\oplus B$. Moreover, there is a natural bijection between group morphisms from $A\oplus B$ and pairs of groups morphisms $A\to C$, $B\to C$:

This bijection is very similar to that for the direct product of sets, above. Each group morphism $f:A\oplus B\to C$ is sent to the pair of group morphisms $f\circ {i}_{1}:A\to C$, $f\circ {i}_{2}:B\to C$.

Examples in $\mathbf{\text{Grp}}$

The quotient group construction

If $N$ is a normal subgroup of a group $G$, then there is a natural bijection between the set of group morphisms $G/N\to H$, and the of group morphisms $G\to H$ with $N$ contained in the kernel. In other words, there is a natural bijection

This bijection requires the "canonical" projection morphism $\pi :G\to G/N$. Using that map, the bijection ${\varphi}_{H}$ sends each group morphism $f:G/N\to H$ to the group morphism $f\circ \pi :G\to H$.

Examples in $R\mathbf{\text{-Mod}}$

The tensor product construction

Suppose $R$ is a commutative ring and $M$ and $N$ are left $R$-modules. By taking the standard $R$-module structure (i.e., $(R,R)$-bimodule structure) on $M$ and the canonical $(R,\mathbf{Z})$-bimodule structure on $N$, we can form the tensor product $M{\otimes}_{R}N$. The result is an $(R,\mathbf{Z})$-bimodule, i.e., a left $R$-module. There is a natural bijection between $R$-module morphisms $M{\otimes}_{S}N\to P$ and certain set maps: