The conversion provided in the image is an example of the Mannich reaction. Let's break down the process and explain why this is the correct answer:

1. Understanding the Reactants and Products:
   The initial structure in the reaction is a pyrrole ring substituted with a methyl group. Formaldehyde (CH₂O) and a secondary amine (dimethylamine, Me₂NH) are used as reagents in the presence of acetic acid (AcOH).
2. Mannich Reaction Explanation:
   The Mannich reaction involves the formation of a β-amino carbonyl compound through the condensation of an aldehyde (or ketone), an amine, and a compound that has an acidic proton (in this case, the pyrrole). Formaldehyde and the amine form an iminium ion which subsequently reacts with the pyrrole to form the final Mannich base.
   The resulting structure is a tertiary amine attached to the carbon adjacent to the nitrogen in the pyrrole, which is characteristic of a Mannich-type reaction.
3. Eliminating Other Options:
   
   • Arndt-Eistert homologation: A method used to elongate the carbon skeleton; not applicable here.
   • Michael addition: A reaction involving the conjugate addition of a carbanion to an α,β-unsaturated carbonyl compound; does not match the transformation.
   • Chichibabin amination reaction: Involves direct amination of heterocycles like pyridine under basic conditions, not applicable here.
4. Conclusion:
   The given reaction fits perfectly with the steps and conditions of the Mannich reaction, confirming the correct option.