By Eugenia L. Cheng

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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.

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