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Every functor $C^n \to C$$F:C^n \to C$ is universal in your sense. Take $I$ to be any connected category and set $Q = F \circ \pi$ where $\pi : C^n \times I \to C^n$ is the projection.
Every functor $C^n \to C$ is universal in your sense. Take $I$ to be any connected category and set $Q = F \circ \pi$ where $\pi : C^n \times I \to C^n$ is the projection.
Every functor $F:C^n \to C$ is universal in your sense. Take $I$ to be any connected category and set $Q = F \circ \pi$ where $\pi : C^n \times I \to C^n$ is the projection.
Every functor $C^n \to C$ is universal in your sense. Take $I$ to be any connected category and set $Q = F \circ \pi$ where $\pi : C^n \times I \to C^n$ is the projection.
