# category theory, $F$ is free object with basis $X$ iff $(F,f)$ is initial object in $D$

Let $$C$$ be a concrete category, let $$F$$ be an object in $$C$$, let $$X$$ be a nonempty set, let $$f : X → F$$ be a function between sets. Let $$D$$ be a category whose objects are pairs $$(B, g)$$, where $$g : X → B$$ are functions between sets, and morphisms from the class $$Hom((B, g),(H, h))$$ are defined as morphisms $$φ: B → H$$ in $$C$$ such that $$φ \circ g = h.$$ Show that $$F$$ is a free object with basis $$X$$ in $$C$$ if and only if $$(F, f)$$ is an initial object in $$D.$$

Definition of free object: Let $$C$$ be a concrete category, let $$F$$ be an object in $$C$$, let $$X$$ be a nonempty set, let $$f : X → F$$ be a function between sets. The object $$F$$ is called free with basis $$X$$ if and only if for each object $$H$$ and each function between sets $$h : X → H$$ there is exactly one morphism $$φ:F→H$$ such that $$φ \circ f = h.$$

Definition of initial object: Let $$C$$ be a category. An object $$I$$ of $$C$$ is called an initial object, if for each object $$T$$ of $$C$$ there is exactly one morphism $$i:I→T.$$

So, my question: is it as easy as I think? I mean both ways, there is nothing to check because this is straight from definition.

• Yes, there isn't much to check other than the definitions. I don't see why the author desires to exclude free objects with basis $\emptyset$, since not only there are notable examples, but it all works exactly the same. – Saucy O'Path May 11 at 16:41