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Let $k$ be a field. We compute the set ${\left[{𝐏}^1,{𝐏}^1\right]}^{\mathrm{N}}$ ofhomotopy classes of pointed $k$-scheme endomorphisms of the projective line ${𝐏}^1$. Our result compares well with Morel’s computation in [11] of the ${\left[{𝐏}^1,{𝐏}^1\right]}^{{𝐀}^1}$ of ${𝐀}^1$-homotopy classes of pointed endomorphisms of ${𝐏}^1$: the set ${\left[{𝐏}^1,{𝐏}^1\right]}^{\mathrm{N}}$ admits an a priori monoid structure such that the canonical map ${\left[{𝐏}^1,{𝐏}^1\right]}^{\mathrm{N}}\to {\left[{𝐏}^1,{𝐏}^1\right]}^{{𝐀}^1}$ is a group completion.