We will talk about the small object argument in this post, that is due to Quillen, and is well suited with respect to the lifting properties. See Section 7.12 in [1].

Let be a category admitting small colimits. Suppose is a functor from a small category to . Then there is a natural map which induces a map for each object . Thus there is a natural map

().

We call is **sequencially small** if the natural map () is an isomorphism.

Recall that in a category , a map is said to have **right lifting property** (RLP) with respect to a map if for every commutative diagram

there is a lifting map . Similarly, is said to have **left lifting property** (LLP) with respect to the map .

In the subsequent, given a family of maps in a category , we are going to show that there is a way of factorizing the map in as such that has the RLP with respect to every map in , provided that each is sequencially small.

**Standard construction.**

Denote by .

Then we define as the pushout of the following diagram

thus there is a map such that .

Inductively, suppose has constructed with such that . Denote by . Then we construct as the pushout of the following diagram

thus there is a map such that .

We denote by , and .

**Proposition**. Suppose each is sequencially small. Then factorizes as such that has the RLP with respect to every map in .

**Proof**. Since each is sequencially small, there is some such that the map factors through . Consider the following diagram

Then the pushout construction from to implies that there is a map making the diagram commute. In particular, composed with the structure map , we deduce that has the desired RLP with respect to the family of maps.

**References**

[1] W. G. Dwyer and J. Spalinski, Homotopy theories and model categories, 1995

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