nachsehen
to look up
To check something by searching for information in a particular place, for example in a book or on the internet.
Ich muss die genaue Adresse im Internet nachsehen.
I need to look up the exact address on the internet.

to look up / check / forgive
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to look up
To check something by searching for information in a particular place, for example in a book or on the internet.
Ich muss die genaue Adresse im Internet nachsehen.
I need to look up the exact address on the internet.

'Nachsehen' is a separable verb. The prefix 'nach-' comes at the end of the main clause. In subordinate clauses and in the infinitive with 'zu', the verb stays together.
Ich sehe schnell die Öffnungszeiten nach.
I'll quickly look up the opening hours.
The case of the object helps you recognize the meaning. 'Etwas (Akkusativ) nachsehen' means to check information. 'Jemandem (Dativ) nachsehen' means to watch a person as they leave. 'Jemandem (Dativ) etwas (Akkusativ) nachsehen' means to forgive a mistake.
Kannst du die Telefonnummer nachsehen? (Information)
Can you look up the phone number? (information)
When it is about checking information, the place where you search is often mentioned. Typical phrases are 'im Wörterbuch nachsehen' (to look up in the dictionary), 'im Internet nachsehen' (to look up on the internet), or 'in den Unterlagen nachsehen' (to check in the documents).
Wenn du das Wort nicht kennst, sieh doch im Wörterbuch nach.
If you don't know the word, just look it up in the dictionary.
In the meaning 'to check whether everything is all right', the preposition 'nach' + dative is often used. Another common construction is a subordinate clause with 'ob' (whether/if).
Der Vater geht nachts oft nach seinem kranken Sohn sehen.
The father often goes to check on his sick son at night.
In the meaning 'to forgive', the verb always needs a person in the dative. This is the person whose mistake you forgive. The mistake itself is in the accusative.
Bitte sieh es mir nach, ich habe es nicht mit Absicht getan.
Please forgive me for it; I didn't do it on purpose.
to see
'nach jemandem oder etwas sehen' and 'nachsehen' can both mean that you check whether everything is all right. 'nach jemandem sehen' sounds more natural when you are taking care of a person or an animal. 'nachsehen' is also used very often when you are looking for information, for example in a book, on the internet, or in a list. Learners may confuse the words because they look similar and partly overlap. Remember: When you are concerned about someone, 'nach jemandem sehen' often fits; for information, 'nachsehen' usually fits.

Herr Weber sieht in der Pause im Wörterbuch nach, wie das Wort geschrieben wird.
During the break, Mr. Weber looks up in the dictionary how the word is spelled.
Ich sehe nach der Nachbarin, weil sie heute noch nicht draußen war.
I am checking on the neighbor because she has not been outside yet today.