Universal Property for isomorphic objects

Question

Suppose $A,B$ are isomorphic objects of some category $C$. Suppose that A satisfies a given universal property does B satisfy the same?

If not can you provide a counterexample for an elementary universal property like say co-product,quotient etc.

Answer 1

If two objects $i$ and $i'$ of a category $\cal C$ are isomorphic, and if $i$ is initial, then so is $i'$.

Note that each arrow $f:a\to b$ in the category $\cal C$ induces a dual map $f^*:\mathcal C(b,c)\to\mathcal C(a,c)$ sending any arrow $g$ to the composite $gf$. This assignment is actually a contravariant functor from $\mathcal C$ to $\mathbf{Set}$ for a fixed object $c$. In particular, if $f$ is an isomorphism $a\cong b$, then so is $f^*$.
If $i$ is initial object, then $|\mathcal C(i,c)|=1$ for each $c$, and this implies that $|\mathcal C(i',c)|=1$ for each $c$ if $i'$ is isomorphic to $i$. Thus any object to an initial object is again initial.

Now a universal property is usually formalized with the notion of a universal arrow. Given a functor $G:\mathcal A\to\mathcal X$, a universal arrow from $x$ to $G$ is a pair $(f,a)$ where $a$ is an object in $\mathcal A$ and $f$ is an arrow $f:x\to Ga$ such that for each arrow $g:x\to Gb$, there is a unique arrow $g':a\to b$ such that $Gg'\circ f = g$, in other words, each arrow $g:x\to Gb$ factors uniquely through $f$.
An example is the functor $\Delta:\mathcal C\to\mathcal C^2$ sending each object $c$ to $(c,c)$. Given a pair $(x,y)$ of objects in $\mathcal C$, the pair of injections $(i_x,i_y):(x,y)\to\Delta(x\coprod y)$ is universal from $(x,y)$ to $\Delta$.

In order to apply the preliminary observation about inital objects, one should express universal arrows as inital objects in a certain category. Indeed, if $(f,a)$ is universal from $x$ to $G:\mathcal A\to\mathcal X$, then this pair is initial in the comma category $(x\downarrow G)$, and vice versa. So if the arrows $(f,a)$ and $(g,b)$ are isomorphic, then both are universal.

Note that it actually suffices that the objects $a$ and $b$ are isomorphic in $\mathcal A$, for if $j:a\to b$ is that isomorphism, then $j:(f,a)\to(G(j)\circ f,b)$ is an isomorphism as well. For example, in $\mathbf{Set}$, if $i_X:X\hookrightarrow X\coprod Y$ and $i_Y:Y\hookrightarrow X\coprod Y$ are the canonical inclusions into the disjoint union, and if the set $Z$ is isomorphic to $X\coprod Y$, then we obtain inclusions $X\hookrightarrow Z$ and $Y\hookrightarrow Z$, and these maps satisfy the universal property of the coproduct.