There are two common formulations of universal properties, which I will briefly state for clarity. Anyone familiar with the definitions of universal element and universal arrow could skip to the bold text.
- Universal Element: Definition 2.3.3 of Rhiel's "Category Theory in Context":
A universal property of an object $c$ in a category $C$ is given by the data of a functor $F \colon C \to \sf Set$ plus a universal element $x \in Fc$, which determines an isomorphism $F\cong C(c,-)$ via Yoneda. A similar definition works for functors $F \colon C^{\rm op} \to \sf Set$.
- Universal Arrow: The first definition on pp. 55 in III.1 of MacLane's "CFTW":
Given categories $C$ and $D$, a functor $F : C \to D$ and an object $d \in D$, a universal arrow from $d$ to $F$ is a pair $(c \in C, u \colon d \to Fc)$ such that for any $f \colon d \to Fc'$ there exists a unique $h \colon c \to c'$ with $Fh \circ u = f$. This is equivalent to saying that $u$ is initial in the category $d \downarrow F$. There is a similar definition of universal arrow from a functor.
I know that universal elements can be seen as a special case of universal arrows. Just take $D = \sf Set$ and $d=1$. Then a universal arrow $u:1 \to Fc$ from $1$ to some functor $F \colon C \to \sf Set$ defines a representation of $F$ thus lends $c$ a universal property. This can be seen as a special case of MacLane's Proposition 1:
An pair $(c, u \colon d \to F c)$ is universal if and only if it defines an isomorphism $D(d,Fc') \cong C(c,c')$ natural in $c'$.
Now, given a universal property in terms of universal elements, translating it directly into the style of universal arrows is somewhat uninteresting since the arrow involved carries no more information than an element. So my question is the following:
Is there a way to translate any universal property in the style of universal elements into the style of universal arrows so that the arrow involved has more information than a mere element?
Let me illustrate what I mean with an example:
Let $V,W \in \sf Vect_\mathbb{K}$ and let $\text{Bilin}(V,W;-) \colon \sf Vect_\mathbb{K} \to Set$ be the functor which sends any $U \in \sf Vect_\mathbb{K}$ to the set of all bilinear maps from $V \times W$ to $U$. Then $$\text{Bilin}(V,W;-) \cong {\sf Vect}_\mathbb{K}(V \otimes_\mathbb{K}W,-).$$ This defines a universal property of $V \otimes_\mathbb{K}W$, with universal element $\otimes \colon V \times W \to V \otimes_\mathbb{K}W$. Now, translating this directly into the language of universal arrows lends that $$(V \otimes_\mathbb{K} W, u \colon 1 \to \text{Bilin}(V,W;V\otimes_\mathbb{K}W))$$ is a universal arrow from $1 \in \sf Set$ to $\text{Bilin}$, where $u(*) = \otimes$.
This is clearly not an interesting reformulation of the original statement. However, I am want to say that $\otimes \colon V \times W \to V \otimes_\mathbb{K} W$ is a universal arrow, since any bilinear $f \colon V \times W \to U$ factors uniquely as $\bar{f} \circ \otimes = f$, where $\bar{f} \colon V\otimes_\mathbb{K} W \to U$ is linear.
EDIT: Thanks to azif00's comment, we can see that $(V \otimes_\mathbb{K}W, \otimes)$ is a universal arrow from $V$ to ${\sf Vect}_\mathbb{K}(W,-)$. So this can just be seen as a special instance of the tensor/hom adjunction.
To rephrase the original universal element style formulation in an intersating way took a little bit of thought here (I have to thank azif00 since I did not think of this), I'm wondering if it is always possible to rephrase a universal element style formulation "in an interesting way" in terms of universal arrows. This is a bit vague, but I'm hoping that the example makes clear what I mean here. Is there a general method for this translation or does it require case by case inspection?
The specific example, and how it was rephrased, seemed to use something particular to the set up, namely that a bilinear map $V \times W \to U$ can be expressed as $V \to {\sf Vect}_\mathbb{K}(W,U)$.