By Eugenia L. Cheng
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Extra resources for Category Theory [Lecture notes]
GB So we have unique h such that ∀x : X GB, Gqx ◦ h = x (). We now show that the outside of the diagram (*) commutes, using the universal property of the product GPY . For each y : Y B, we have the following diagram: X f k GY h Gd Gy Ge GQX GPY () Gpy () GB GqGy◦f Now, the outside commutes by (), and the triangles commute as shown. So we need show that Ge ◦ h = Gd ◦ f. We use the universal property of the product GPY to induce a unique k such that for all B, Gpy ◦ k = Gy ◦ f; then we show that Ge ◦ h and Gd ◦ f both satisfy this y: Y condition.
So: P : (X ↓ G) D creates small limits. (X ↓ G) be a diagram. We need to show that, if PD has a limit cone, then Let D : މ there is a cone cI (V DI)I∈މ PcI in (X ↓ G) such that (PV PDI)I∈ މis a limit for PD in D, and that any such cone is itself a limit for D in (X ↓ G). Suppose PD : މ D has a limit cone, say (L cI PDI)I∈ މ: L cI PDI G preserves small limits, so (GL GcI cI PDI cI PDI GPDI)I∈ މis a limit for GPD in C. GL GcI GPDI GcI GPDI GcI GPDI (DI)I∈ މgives a diagram in (X ↓ G) X GPDI GPDI GPDI GPDI)I∈ މin C.
Suppose we have a limit cone for D, ( I DI kI DI)I∈މ HkI We need to show that ( (ރ, I DI) (ރ, DI))I∈ މ. is a limit for H• ◦ D. e. HC ◦D. But we have already shown this, since representables preserve limits, and the given cone is just HC of ( I DI DI)I∈ މ. () Suppose F : ފ × މ have a functor D is such that FJ : މ I D has a limit F(I, ) : J such that J I F(I, J) ∼ = I I F(I, J) for all J ∈ ފ. Then we F(I, J) (I,J) F(I, J) in the sense that if one exists, then so does the other, and they are isomorphic with corresponding limit cones.