Let $({\cal C},\otimes)$ be a monoidal category, $X$ an object in ${\cal C}$, and $\Psi:X \otimes X \to X \otimes X$ an isomorphism such that $\Psi$ satisfies the braid relation:
$$
(\Psi \otimes \text{id}) \circ (\text{id} \otimes \Psi) \circ (\Psi \otimes \text{id}) = (\text{id} \otimes \Psi) \circ (\Psi \otimes \text{id}) \circ (\text{id} \otimes \Psi).
$$
What would one call such an isomorphism? The most obvious suggestion is to call it a *braiding for $X$*. Might this be taken to imply that $\Psi$ comes from a braiding for the category (which I do not want to assume)? 



Name for an Isomorphism in a Monoidal Category that Satisfies the Braid Relation
Assuming S. Carnahan's surmise in his comment is correct, I believe the correct term for this is "Yang-Baxter" operator in a monoidal category (or, you could call an object $X$ equipped with such an automorphism $R: X \otimes X \to X \otimes X$ a [Yang-Baxter object](http://ncatlab.org/nlab/show/braid+category)). This terminology is given in the seminal paper on the subject, Braided Tensor Categories by Joyal and Street (Adv. Math. 102, pp. 20-78, 1993). 


In particular, as observed by Joyal and Street, the braid category can be characterized as the free (i.e., initial in a 2-categorical sense) monoidal category equipped with a Yang-Baxter object. 


**Edit:** Another reference for this terminology: 


* André Joyal and Ross Street, Tortile Yang-Baxter operators in a tensor category, J. Pure Appl. Alg. 71 (1991), 43-51.


