# Initial Object We define initial and terminal objects, which we shall use for describing algebras and initial algebras. (Definition) Let `𝒞` be a category. An object `0` is the initial object of `𝒞` if, for all objects `a`, there is a unique morphism `0 → a`. (Definition) Let `𝒞` be a category. An object `1` is the terminal object of `𝒞` if, for all objects `a`, there is a unique morphism `a → 1`. (Lemma) Initial and terminal objects are unique up to isomorphism. (Proof) Let 𝒞 be a category with initial objects 0 and 0′. There are unique morphisms `00′ ∶ 0 → 0′` and `0′0 ∶ 0′ → 0`, and `00 = id0` and `0′ 0′ = id0′`. Hence, `0′0 ∘ 00′ = id0` and `00′ ∘ 0′0 = id0′`. That is, 0 is unique up to isomorphism. Similarly, let `𝒞` be a category with terminal objects 1 and 1′. There are unique morphisms `11′ ∶ 1′ → 1` and `1′1 ∶ 1 → 1′`, and `11 = id1` and `1′1′ = id1′`. Hence, `11′ ∘ 1′1 = id1` and `1′1 ∘ 11′ = id1′`. That is, 1 is unique up to isomorphism. In Hask, Void (the empty or uninhabited type) is the initial object, and unit or `()` type is the terminal object. In Set, the elements of a set 𝐴 can be considered as functions from a terminal object, that is, any singleton set, to 𝐴. More specifically, if 𝑥 ∈ 𝐴 and 1 is a terminal object, then 𝑥 can be considered as a function 𝑥 ∶ 1 → 𝐴 which assigns 𝑥 to the element of 1. --- # Agent Instructions: Querying This Documentation If you need additional information that is not directly available in this page, you can query the documentation dynamically by asking a question. Perform an HTTP GET request on the current page URL with the `ask` query parameter: ``` GET https://mandober.gitbook.io/math-debrief/450-category-theory/03-key-concepts/initial-object.md?ask= ``` The question should be specific, self-contained, and written in natural language. The response will contain a direct answer to the question and relevant excerpts and sources from the documentation. Use this mechanism when the answer is not explicitly present in the current page, you need clarification or additional context, or you want to retrieve related documentation sections.