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This question deals with certain morphisms and 2-morphisms in the definition of coslice categories in 2-categories. Though I'm aware of the notion of lax (co)slice categories, I believe the following problem is distinct.

Let $\mathcal{C}$ be a 2-category and $X$ be an object of $\mathcal{C}$. The coslice category $X \downarrow \mathcal{C}$ has morphisms $f\colon X \to A$ for objects, and has for morphisms $N\colon (f\colon X \to A) \to (f'\colon X \to B)$ the morphisms $N\colon A \to B$ such that $f'=Nf$.

Assume now that we have $N_1\colon A \to B$, $N_2\colon A \to B$, with $f'=Nf$, $f''=Nf$, and that we also have a natural transformation $\alpha\colon N_1 \to N_2$. Intuitively, this would define some kind of morphism between the objects $f'\colon X \to B$ and $f''\colon X \to B$, but this is not the definition of 1-morphisms in $X \downarrow \mathcal{C}$.

The question is: what extra structure is brought to the category $X \downarrow \mathcal{C}$ by the natural transformations $\alpha\colon N_1 \to N_2$ ? I am thinking that $X \downarrow \mathcal{C}$ might be a double category, but can't quite define it precisely.

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  • $\begingroup$ I'm not sure that this is entirely the right notion for the coslice 2-category. It depends on whether you want it to represent a weak of strong 2-limit. Try ncatlab.org/nlab/show/slice+2-category for a description of the dual notion in the weaker setting. $\endgroup$ – Tyrone Jul 12 '17 at 13:13
  • $\begingroup$ A natural candidate for a $2$-cell in $X \downarrow \mathcal{C}$ from $N_1 : A \to B$ to $N_2 : A \to B$ would be a $2$-cell $\alpha : N_1 \Rightarrow N_2$ in $\mathcal{C}$ such that $\alpha \cdot f = \mathrm{id}_{f'}$, where $\alpha \cdot f$ is an instance of whiskering. $$~$$ [P.S. since $2$-cells must be between parallel $1$-cells, your $f'$ and $f''$ should be equal, and $N_1,N_2$ are both $1$-cells in $X \downarrow \mathcal{C}$ from $f$ to $f'$.] $\endgroup$ – Clive Newstead Jul 12 '17 at 16:06

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