In reading about model theory (David Marker's Book), a witness has a specific meaning, i.e. an element $t$ making $\exists x\ \phi (x)$ true. Some theories have the the witness property. In reading Category Theory In Context, Riehl writes (p 50)
The universal element witnessing the universal property of the complete graph is an $n$-coloring $K_n$, an element of the set $n\text{-}\mathrm{Color}(K_n)$.
which has a clearly related meaning, although it doesn't seem obvious that there is some formula being witnessed. My question is that I am wondering if there is a formal idea here, or if witness just means in effect "makes true". I don't mean to be pedantic, the meaning is understood, and sorry if this is too vague.
Edit: Martin Brandenburg pointed out in the comments that it is clearly a formula, be it in a language I am not familiar with. I think my question more accurately stated is whether the informal usage of "witness" always coincides with its formal usage?
