entsprechend
corresponding
As an adjective: suitable, belonging to something, or appropriate to something.
Bitte wählen Sie die entsprechende Option im Menü aus.
Please select the corresponding option in the menu.

corresponding / according
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corresponding
As an adjective: suitable, belonging to something, or appropriate to something.
Bitte wählen Sie die entsprechende Option im Menü aus.
Please select the corresponding option in the menu.

'Entsprechend' is the present participle of the verb 'entsprechen' (to correspond to). Therefore it can be used as an adjective and retains the meaning of 'suitable' or 'matching'.
Seine Leistung entsprach nicht den Erwartungen.
His performance did not meet the expectations.
When 'entsprechend' is used as a preposition, it always takes the dative. It can stand before or after the noun it refers to.
Entsprechend dem Vertrag müssen wir die Ware bis Freitag liefern.
According to the contract, we must deliver the goods by Friday.
As an adverb, 'entsprechend' often stands at the beginning of a sentence and connects it to previous information. In this position, it triggers inversion, so the verb moves to position 2.
Das Wetter war schlecht. Entsprechend hatten wir schlechte Laune.
The weather was bad. Accordingly, we were in a bad mood.
to / after / according to
'nach' and 'entsprechend' can both mean that something follows a rule, plan, or requirement. 'nach' is very common in everyday language and sounds simple. 'entsprechend' sounds somewhat more formal and precise, especially when rules, regulations, or agreements are involved. So both fit in similar situations, but 'entsprechend' more strongly emphasizes agreement with something. In official or professional contexts, 'entsprechend' is often more natural; in everyday language, 'nach' often is.

Entsprechend den Regeln müssen alle Gäste sich zuerst anmelden.
According to the rules, all guests must register first.
Wir bauen das Regal nach der Anleitung auf.
We are assembling the shelf according to the instructions.