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Initial and terminal objects
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<p>In <a href="/facts/Category_theory/6zS3DXPa">category theory</a>, a branch of <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, an initial object of a <a href="/facts/Category_(mathematics)/7WzxUThf">category</a> C is an object I in C such that for every object X in C, there exists precisely one <a href="/facts/Morphism/Kdpuyz3S">morphism</a> <i>I</i> → <i>X</i>. </p><p>The <a href="/facts/Dual_(category_theory)/frtUrhBq">dual</a> notion is that of a terminal object (also called terminal element): T is terminal if for every object X in C there exists exactly one morphism <i>X</i> → <i>T</i>. Initial objects are also called coterminal or universal, and terminal objects are also called final. </p><p>If an object is both initial and terminal, it is called a zero object or null object. A pointed category is one with a zero object. </p><p>A <a href="/facts/Strict_initial_object/TskUV8xw">strict initial object</a> I is one for which every morphism into I is an <a href="/facts/Isomorphism/pj8KgkxU">isomorphism</a>. </p>